Vibrating String Equation Research Articles

Solution of some half-strip problems in quadratures for the string vibration equation

Solution of some half-strip problems in quadratures for the string vibration equation

Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures

Solution of Some Problems for the String Vibration Equation in a Half-Strip by Quadratures

Innovation and Development of Pipa Art in Contemporary Plucked Instrument Performance

Abstract This study summarizes the direction of innovation in contemporary pipa performance techniques through theoretical analysis of pipa history, performance techniques, form, and production materials. The independent variable of string vibration during pipa performance is set, and the string vibration equation of pipa performance is deduced based on the actual vibration situation. Afterward, we collected and pre-processed the sound signals of the pipa performance and selected simulation experiments to analyze the string vibration and acoustic signals during the pipa performance. The results show that when the plucking force of 1 string is 0.8N, 1.6N, 2.4N, and 3.2N, the corresponding Y-direction displacements are 2.95mm, 6.37mm, 8.85mm, and 12.48mm, with the ratio of 1:2.16:3:4.23, and the maximum Y-direction displacements are monotonically and linearly positively correlated with the plucking force. In addition, the sound power of the two pipas gradually increased with the increase in frequency, which led to a dynamic range of more than 18.53 Hz when the frequency exceeded 8000 Hz. In this study, through the simulation of string vibration and acoustic signals of the pipa playing innovative techniques, the laws and characteristics of the pipa art can be recognized so as to strengthen the awareness of innovation and development of pipa playing and to make the art of pipa better developed and inherited.

Research on the Integration Teaching Mode of Traditional Music Elements and College Piano under the Background of Information Technology

Abstract With the deep integration of information technology and music education, how to integrate traditional music to innovate college piano teaching mode has become a problem that must be solved for the development of the times. Based on the scaffolding teaching method, this paper integrates traditional music elements with college piano in-depth and realizes the innovation of college piano integration teaching mode by adopting online pre-study before class, offline teaching during class and timely reflection and evaluation after class. In the teaching process, traditional music signals are represented by time-frequency domain features so as to realize the digital pre-processing of traditional music signals, and traditional music fundamental tone contours are transformed by using the fundamental tone feature extraction method. Then, based on the principles of physics, we explored the string vibration equation of the piano-generating principle and constructed a mathematical model for the piano music signal. To verify the effectiveness of this paper’s method in piano teaching in colleges and universities, relevant algorithms and teaching practices are used. The results show that the GNSDR of this paper’s method on vocals is improved by 6.58% compared with Unet, and its SDR is 4.25dB and 5.21dB when separating the two music source signals of Piano and Vocals, which is improved by 0.36dB and 0.14dB compared with SHN-4, respectively. The college piano scaffolding teaching model that integrates traditional music elements has enhanced sight-reading ability by 116.44%. Under the background of information technology, it is necessary to fully combine the relevant technology to extract the characteristics of traditional music elements so as to help teachers better carry out the design of piano music teaching programs and improve the quality of colleges and universities to cultivate piano talents who inherit traditional music.

A non-local problem with integral conditions of the first kind for the string vibration equation

A non-local problem with integral conditions of the first kind for the string vibration equation

Perceptual and cognitive infiltration of piano lessons for college students under the concept of digital health

Abstract In this paper, firstly, according to mental health education, the research on the perception penetration of the piano course is illustrated from the two directions of body perception and mind perception. Secondly, it researches the music feature extraction in the piano course, obtains the amplitude, wavelength and frequency of the music feature according to the piano string vibration equation, then applies the short-time Fourier analysis to identify the music feature and reconstructs the piano timbre by using the identification result, which provides technical support for the design and realization of the piano course teaching system. Then, the overall design direction of the piano course teaching system is determined according to the teaching needs of the piano course in colleges and universities, and then the core module of the piano course system is realized. At the same time, the teaching of piano courses is empirically analyzed through statistical analysis. The results show that there is a significant difference between piano course teaching and the degree of perception and cognition in piano course teaching (F=4.194, p<0.05), there is a significant difference between piano course teaching and the degree of cognition (F=4.243, p<0.05), and there is a significant difference between piano course teaching and interpersonal relationship (F=5.391, p<0.05). This study not only has a promoting effect on piano teaching in colleges and universities but also improves the guiding value of piano curriculum reform in colleges and universities.

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Fourier series and its property

Fourier analysis appeared to solve some partial differential equations at first. In 1747, D’ Alembert solved the vibrating string equation using the method of traveling waves. In 1753, D. Bernoulli proposed the solution which for all intents and purposes is the Fourier series, but Euler did not convince of its full generality entirely since he could not make sure whether any function could be expanded in the Fourier seriesFourier analysis is an important theorem of modern mathematics. It deeply influences the development of partial differential equations and information science. This paper will talk about some basic propositions of the Fourier series and some simple theorems of the Fourier transform. It will use an algebraic view to talk about convolution and good kernels. It will refer to some methods of Hilbert space and complex analysis.

High-accuracy estimation method of typhoon storm surge disaster loss under small sample conditions by information diffusion model coupled with machine learning models

Typhoon storm surge disaster is the most severe marine disaster in China. Accurate estimation of typhoon storm surge disaster loss (TSSDL) is significant for emergency decisions and economic sustainability development. However, the TSSDL estimation is limited by small sample conditions, resulting in the low accuracy of the TSSDL estimation model based on machine learning. To solve the problems of easy overfitting and poor generalization ability of machine learning models under small sample conditions, the high-accuracy TSSDL estimation method was proposed. Firstly, this estimation method combines Gaussian Noise with the Information Diffusion Model based on the Vibrating String equation (named GN-VSIDM) to generate virtual samples to augment the original training set. Then, the augmented training set was applied to train three machine learning models. The results are as follows: the virtual sample generation method, i.e., GN-VSIDM, solves the small sample problem in the TSSDL estimation process and improves the machine models’ accuracy and robustness. Based on the GN-VSIDM and the eXtreme Gradient Boosting (named XGBoost) methods, the joint model GN-VSIDM-XGBoost is the optimal TSSDL estimation model. Compared with the original XGBoost model, the RMSE and R2 of the GN-VSIDM-XGBoost model are 0.1089 and 0.8292, which reduces 25.67% and improves 19.88%, respectively. Besides, the GN-VSIDM-XGBoost model possesses excellent robustness. The GN-VSIDM overcomes the limitation on the performance of TSSDL estimation models based on machine learning under small sample conditions. This study provides an effective case and method for solving the small sample problem in disaster loss assessment.

Free Response and Musical Pitch of Shape Memory Alloy Wires Under Voltage Loading

An experimental setup, on which the shape memory effect is based, is developed to measure the free response frequency and musical pitches of SMA wires. In this paper, the voltage, force, and sound signals of SMA wires are determined simultaneously under two kinds of experiments, constant temperature uniaxial tensile tests and fixed-length thermal cycle tests. Based on constitutive equations and thermodynamic equations, an empirical formula for the driving voltage and the steady-state stress of SMA wires is proposed, and the undetermined coefficients at radii of 0.050 mm and 0.070 mm are determined by experiments. The musical pitches of SMA wires under a constant voltage are obtained by combining the empirical formula with the string vibration equation. The theoretical results are compared with the actual musical pitches measured by the sound signals, and the two have a high degree of coincidence. Just for an SMA wire with a radius of 0.070 mm, the free response frequency range reaches 640 Hz to 1120 Hz, which covers 6 whole musical tones in total. Thus the SMA wire can be used as a string to play simple music.

On Problem of Internal Boundary Control for String Vibration Equation

The article deals with the vibration control problem described by one dimensional wave equation with integral type boundary condition. As usual, the initial and final moments of time for arbitrary displacements and velocities of the wave are specified by points on a string (Cauchy data). It is shown that the minimum time for the realizable control is uniquely determined by the condition of correct solvability to the Cauchy problem involving data lying on disconnected manifold. This suggests that the internal boundary conditions does not affect the minimum time value. Necessary and sufficient conditions for the existence of the desired internal-boundary controls that move the process from the state initially specified to a predetermined final one are obtained and written out. The controls are presented in explicit analytical form. Moreover, it is shown that for the inner-boundary controls expressions, one should use not the representation of the solution to the Cauchy problem in the sought-for domain, but the formula for the general solution of the string oscillation equation (d’Alembert’s formula).

On one boundary control problem of string vibrations with given velocity of points at an intermediate moment of time

The boundary control problem for the string vibration equation with given initial and final conditions, with the given velocity of the string points at an intermediate moment of time is considered. It is controlled by displacement of one end while the other end is fixed. A constructive approach for constructing the boundary control action is proposed, which is applied for given functions.

Bridge Dynamic Cable-Tension Estimation with Interferometric Radar and APES-Based Time-Frequency Analysis

Dynamic cable-tension is an important bridge-health indicator. However, it is difficult to be measured precisely and efficiently. A remote bridge dynamic cable-tension measurement method is proposed. It uses an interferometric radar sensor, a time-frequency analysis technique, and a tension estimation approach based on a string-vibration-equation. One radar can measure the displacements of multiple cables aligned on one side of a bridge, at the same time. By solving the string vibration equation, each cable-tension is calculated from its fundamental frequency, which is obtained by time-frequency analyzing a short section of the cable’s whole displacement vector in an overlapped-piecewise manner. An adaptive amplitude and phase estimation (APES) algorithm is used to solve the frequency resolution deterioration problem due to the short duration. Simulations and field experiments with a K band interferometric radar validate that the proposed method is superior to traditional cable-tension measurements in terms of precision, robustness, and efficiency. The proposed method is of great application value in measuring and monitoring large cable-stayed bridges and cable-suspended bridges.

On the Ergodic Theory of Equations of Mathematical Physics

Linear evolution equations of mathematical physics admitting an invariant in the form of a positive quadratic form are considered. In particular, this includes the string vibration equation, the Liouville kinetic equation, the Maxwell system of equations and the Schrodinger equation. Conditions for the existence of an invariant Gaussian measure are indicated, which makes it possible to apply well-known results of ergodic theory (Poincare’s recurrence theorem, Birkhoff–Khinchin ergodic theorem, etc.). We discuss the Hamiltonian property of such systems and conditions for their complete integrability. The ergodic properties of Kronecker flows on infinite-dimensional tori are studied. A general theorem on the averaging of quadratic forms is established.

Matching conditions for values of characteristic oblique derivative at the end of a string, initial data and right-hand side of the wave equation

Sufficient matching conditions the time-dependent characteristic first derivatives in the boundary mode with the initial conditions and the more general vibration equation of a semi-bounded string are derived in the sets of solutions of all higher order smoothness orders. They generalize the previously found sufficient matching conditions in the case of a similar mixed problem for the simplest string vibration equation. The characteristic of non-stationary first oblique derivatives in the boundary mode means that at each moment of time they are directed along the critical characteristic.

Optimal Control of String Vibrations with Nonseparate State Function Conditions at Given Intermediate Instants

An optimal control problem for the string vibration equation with given initial and terminal conditions, nonseparated states at intermediate instants and a performance criterion defined on the entire time interval is considered. This problem is solved using the methods of separation of variables and the theory of optimal control of finite-dimensional systems with nonseparated multipoint intermediate conditions. As an illustrative example of the approach proposed in the paper, an optimal control of string vibrations with given nonlocal deflection values of the string at some intermediate instants is constructed.

Задача граничного управления колебаниями струны смещением левого конца при закрепленном правом конце с заданными значениями функции прогибав промежуточные моменты времени

We consider the boundary control problem for the homogeneous string vibrationequation with given the classical boundary (initial and final) conditions and with given valuesof the deflection function at intermediate times. The control is performed by displacementof the left end of the string when the right end is fixed. The problem is reduced to thecontrol problem with zero boundary conditions. We propose the constructive method forconstructing the boundary control of the process of string vibrations with given values ofthe deflection function at intermediate times.We present the results of numerical experimentsand the corresponding graphs confirm the validity of the results.

Quasi hyperbolic function expansion method and tanh-function method for solving vibrating string equation and elastic rod equation

This paper applies the quasi hyperbolic function expansion method and the modified extended tanh-function method to exactly solve the nonlinear vibrating string equation and the elastic rod equation. The obtained exact solutions can be expressed by hyperbolic functions, which can outline various vibration properties.

On a Generalization of the Initial-Boundary Problem for the Vibrating String Equation

In the present paper, we study a generalization of the initial-boundary problem for the inhomogeneous vibrating string equation. The initial conditions include the higher order derivatives of the unknown function. The problem is studied under homogeneous boundary conditions of the first kind. The uniqueness and existence of a regular solution of the problem are proved. To prove the main result we use the spectral decomposition method.

Estimation of Equation Coefficients of Free and Forced String Vibrations in Continuous Medium with Consideration of Dynamic Noise and Measurement Noise

Abstract Linear hyperbolic differential equation of partial derivatives, that sets string vibration, as well as other oscillatory processes in continuous medium and electrodynamics, is also used in many areas of theoretical physics. This article considers estimation problems of equation coefficients of free and forced string vibrations with fixed ends with consideration of additive dynamic and measurement noises. Use of a finite-difference method allowed solving partial differential equations with consideration of additive dynamic and measurement noise by solving systems of linear stationary stochastic, finite-difference and algebraic equations, expressed as state-space models. State-space model representation of string vibration equation makes it possible to use Kalman filter algorithm to make accurate estimates of string vibrations’ coordinates.